Template:ZPI

Template documentation[view] [edit] [history] [purge]

This template should not be substituted.

This template generates a table for a particular edo's zeta peak index.

Parameters

All parameters are required.

{{{steps}}}

The zeta-optimised number of steps per octave, corresponding to the location of a particular edo's nearest local maximum or minimum according to the Riemann–Siegel Z function,

$$ Z\left( \frac{2\pi}{\ln 2} \, x \right) $$

For example, 12edo's local zeta-optimised step count is 12.0231830072926.

{{{step size}}}

The step size in cents, which is 1200 divided by the aformentioned zeta-optimised step count.

For the height of the zeta peak, you can use tempered height and pure height or just height.

{{{tempered height}}}

The absolute value of Z evaluated at the zeta-optimised step count. For 12edo, this value is 5.193290 when evaluated at 12.0231830072926.

Must be used in conjunction with pure height, do not use with height.

{{{pure height}}}

The absolute value of Z evaluated at the exact integer. For 12edo, this value is 5.084467 when evaluated at exactly 12.

Must be used in conjunction with tempered height, do not use with height.

{{{height}}}

The absolute value of Z evaluated at a ZPI that does not correspond strongly to an edo. Do not use with tempered height or pure height.

{{{integral}}}

The absolute value of the integral between the two roots (zeroes) of Z closest to the given edo. For 12edo, this value is 1.269599.

{{{gap}}}

{{{edo}}} (optional)

This parameter is optional. Pass in the edo that the ZPI represents if the page's title does not contain it.

{{{octave}}}

The zeta-optimised octave size. For 12edo, this is 1197.68616940005.

{{{consistent}}}

The ZPI's consistency limit.

{{{distinct}}}

The ZPI's distinct consistency limit.

Example (12edo)

Here is the complete zeta peak index table for 12edo as it appears on the page:

{{ZPI
| zpi = 34
| steps = 12.0231830072926
| step size = 99.8071807833375
| tempered height = 5.193290
| pure height = 5.084467
| integral = 1.269599
| gap = 15.899282
| octave = 1197.68616940005
| consistent = 10
| distinct = 6
}}

Tuning			Strength				Octave (cents)		Integer limit
ZPI	Steps per 8ve	Step size (cents)	Height		Integral	Gap	Size	Stretch	Consistent	Distinct
			Tempered	Pure
34zpi	12.023183	99.807181	5.19329	5.084467	1.269599	15.899282	1197.686169	−2.313831	10	6